Structured Recurrent Models Recurrent models have been widely adopted to tree data by adapting the recurrent cell update to the given structure. The most notable example in this setting is given by Recursive Neural Networks [1]}, in which the recurrent cell state is updated recursively by navigating the tree structure bottom-up, i.e. from leaves to root. Subsequently, several model variants have been proposed in order to define more efficient and scalable state update methods. The most notable example is given by Tree-LSTM[2]}. The main drawback of such models is that of handling binary tree structures only. Such limitation drastically impacts on the nature of the inputIt is always possible to retrieve an equivalent binary tree in terms of node ordering. However, the tree structure significantly changes.. To this end, several solutions, such as Tree-Net [3]}, ARTree-LSTM [4]} and CP-LSTM [5]}, propose novel state update mechanisms that extend the model to any n-ary tree structure. Albeit our experimental scenario focuses on tree structured data only, our regularization methodology can be extended to any kind of domain (with ad hoc constraints) and to graph structures as well. For instance, we can consider TC-GNNs for graphs while imposing node clusters to define a DAG, similarly to [6]}. On the other hand, the aforementioned recurrent models are entirely centred on tree structured data. Regarding applications of GNNs to learn tree structured data, the closest examples to our experimental setting are [7]} and [8]}. In the first, authors apply tree structured templates for node aggregation by exploiting domain specific entity relations. Such approach could be potentially be implemented as a tree regularized pooling operation where node clusters are enforced to match domain specific templates. Conversely, in [8]} a high level tree structure, denoted as H-tree is defined from a starting graph by applying the junction tree decomposition algorithm [10]} in order to achieve efficient message-passing. Such method is particularly important when handling large complex graphs. On the other hand, we are mainly interested in extracting few key structured features from input data for efficient and sparse tree representation and, potentially, tree comparison.
